Dimension of the vector space of odd polynomials of degree $n$

I was doing some exercises of vector space. Don't know how to handle this question?

A polynomial $p(x)$ is odd if $p(-x)=-p(x)$. Let $O_n$ denote the sets of odd polynomials in $P_n$. Then what is $\dim\,O_n$?

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Do you know what the dimension of $P_n$ is, and how to show it? Which monomials are in $O_n$? Do you know the criterion on the coefficients of a polynomial $p(x)=a_0+a_1x+a_2x^2+\cdots+a_{n-1}x^{n-1}+a_nx^n$ for $p(x)$ to be in $O_n$? –  Jonas Meyer Jan 29 '12 at 5:17

Remember, a polynomial P(x) is odd iff it only contains odd powers of x. So every odd polynomial is of the form:

$a_1x^1 + a_3x^3 + \cdots + a_kx^k$

Assuming $P_n$ means the vector space of polynomials of degree n, you can figure out the number of different coefficients $a_i$ that will show up for a given value of n without too much trouble. That number will be the number of "degrees of freedom" you have in choosing an odd polynomial (equivalently, the number of basis elements), so it will be the dimension of the vector space.

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I assume $P_n$ means the vector space of polynomials of degree $\le n$ rather than the set (and non-vector space) of polynomials of degree $=n$, and similar for $O_n$. –  anon Jan 29 '12 at 5:37
anon: Right, thanks, corrected. @JavaMan: I wasn't intending to worry about the values of a_i, just how many there are of them. I hope the answer isn't written confusingly. –  Lopsy Jan 29 '12 at 5:40

We can split functions into odd and even parts via $$f_{\mathrm{odd}}(x)=\frac{f(x)-f(-x)}{2},\qquad f_{\mathrm{even}}(x)=\frac{f(x)+f(-x)}{2}.$$

1: Check that these are odd and even respectively, and that $f=f_{\mathrm{odd}}+f_{\mathrm{even}}$.

2: Check that $(f+g)_{\mathrm{odd}}=f_{\mathrm{odd}}+g_{\mathrm{odd}}$ and $(f+g)_{\mathrm{even}}=f_{\mathrm{even}}+g_{\mathrm{even}}$.

3: Show that $f$ is odd (resp. even) if and only if $f_{\mathrm{even}}$ (resp. $f_{\mathrm{odd}}$) is identically zero.

4: Observe that even powers $x^{2k}$ are even and odd powers $x^{2k-1}$ are odd.

5: Explain why $f\in P_n$ is odd (resp. even) if and only if it contains no even (resp. odd) powers.

6: Observe that $\{x,x^3,x^5,\dots,x^m\}$ is a basis for $O_n$, where $m$ is the largest odd $\le n$.

Now how many elements are in this set?

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$dimO_n=n/2$ if n is even and $dimO_n=(n+1)/2$ if n is odd. Thank you. Its very helpful. Why are you doing step 1, 2 and 3? Are you checking $O_n$ and $E_n$ are subspaces of $P_n$? –  Shannon Jan 29 '12 at 23:48
Shannon: Step 1 checks that the given expressions are in fact the odd and even parts. Step 2 and 3 would help prove 4 if needed. –  anon Jan 29 '12 at 23:50